Kane’s Equations for Haptic Display of Multibody Systems

R. Brent Gillespie

Haptics-e, The Electronic Journal for Haptics Research, Vol 2., No. 3., 18 August 2003, http://www.haptics-e.org

The design of a general-purpose multibody simulator that runs in real-time and features haptic display is presented. The repertoire of this simulator includes systems subject to holonomic constraints, nonholonomic constraints, and notably, systems subject to changing constraints. In contrast to the use of generic equations of motion, Kane’s method is used by a computerized symbolic manipulator to produce custom-built, compact, and computationally efficient ordinary differential equations for each virtual system to be simulated. A method is presented whereby new equations, reflecting the presence of a transient constraint, may be formulated on-line by a recombination of terms comprising the nominal equations. This paper also serves as a tutorial introduction to Kane’s method and includes a detailed derivation of Kane’s equations with a geometric interpretation. An illustrative  example featuring changing constraint conditions is developed in the body of the paper while two more examples, chosen to highlight certain features within Kane’s method, are presented at the conclusion.